Problem 48 - Intermediate Algebra

Question

Divide, and then simplify, if possible. See Objective 3. $$ \frac{17}{12} \div \frac{34}{3} $$

Step-by-Step Solution

Answer

The simplified result of \(\frac{17}{12} \div \frac{34}{3}\) is \(\frac{1}{8}\).

Step 1: Understand the Division of Fractions

The division of two fractions \(a/b \div c/d\) is equivalent to multiplying the first fraction by the reciprocal of the second fraction. This means we need to flip the second fraction (\(c/d\) becomes \(d/c\)) and then multiply.

Step 2: Apply the Reciprocal and Multiply

First, write the division problem as a multiplication problem using the reciprocal of the divisor \(\frac{34}{3}\). This gives us: \(\frac{17}{12} \times \frac{3}{34}\).

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{17 \times 3}{12 \times 34}\). This results in \(\frac{51}{408}\).

Step 4: Simplify the Resulting Fraction

Find the greatest common divisor (GCD) of 51 and 408, which is 51. Divide both the numerator and the denominator by 51: \(\frac{51 \div 51}{408 \div 51}\), which simplifies to \(\frac{1}{8}\).

Step 5: Verify Simplification

Check to ensure that \(\frac{1}{8}\) cannot be simplified further. Since 1 and 8 have no common factors other than 1, this is the simplest form.

Key Concepts

Reciprocal of a Fraction

When dividing fractions, one essential concept to understand is the reciprocal. A reciprocal of a fraction is created by swapping its numerator and denominator. For example, the reciprocal of \( \frac{34}{3} \) is \( \frac{3}{34} \). You simply flip the two numbers.

This idea is key because dividing by a fraction is the same as multiplying by its reciprocal. So, instead of dividing \( \frac{17}{12} \) by \( \frac{34}{3} \), you multiply \( \frac{17}{12} \) by \( \frac{3}{34} \).

Reciprocals are not only used in division problems but also play a role in equations and other mathematical operations. Knowing how to find a reciprocal is a fundamental skill in math.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and the denominator have no common factors other than 1. This makes them easier to work with and recognize as equivalent to any fractions with larger numbers.

To simplify \( \frac{51}{408} \), we find numbers that can divide both the top and bottom evenly. This process ensures the fraction is in its simplest form. First, identify the Greatest Common Divisor (explained later). Divide both the numerator and the denominator by this number. In this case, dividing both by 51 gives us \( \frac{1}{8} \).

• This step makes comparing, adding, or subtracting fractions cleaner.
• Helps in visualizing and understanding numbers better.

Always ensure, after simplifying, that no further simplification is possible.

Greatest Common Divisor

The Greatest Common Divisor (GCD), sometimes known as the greatest common factor, is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. This is crucial for simplifying fractions.

To find the GCD, list the factors of both numbers and determine the largest factor they have in common. For instance, in the fraction \( \frac{51}{408} \), both 51 and 408 are divided perfectly by 51, making it the GCD.

• Use the GCD to simplify fractions efficiently.
• Aids in the simplification of algebraic expressions, reducing errors.

Knowing how to calculate the GCD is a valuable skill in math that helps in various contexts beyond just fractions.